Full civil registration of births, marriages and deaths started in 1864, anything before that time would require a search of Church records - e.g. baptisms. Just to mention that surname spelling was not fixed - so could be either Johnston or Johnson
Do you know what religious denomination your Johnston/Graham family were ?
A search of Catholic baptism records is not showing any promising matches for the children for me at the moment...Church of Ireland baptisms are not as well represented as Catholic online at the moment.
I checked the 1887 Robert Johnston / Sarah Graham marriage - it took place in a Church of Ireland, so may explain the lack of baptism results. Non-Catholic marriages (i.e. Protestant & Registry Office) were registered from 1845 so If this family were Church of Ireland the marriage of James & Jane should be recorded in civil records - but I'm not seeing a good match...
The only close match, by date and location, is one in 1858 in Cootehill district of a James Johnston and Jane Grimes
The marriage took place in a Church of Ireland Church in Cootehill registration district, I cant make out the church or parish yet, will update ...
Both the bride and groom 'signed' with their marks so probably illiterate.. Graham and Grimes could be mixed up or maybe misheard, the groom's occupation fits - labourer, other than the possible bride's surname mixup seems a reasonable fit
I check further and also see if I can locate any further civil births after 1864 and update here..